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Numerical estimates for the chemicalcomposition of Enceladus’ plume particles
Onni Vetelainen
Supervisors: Nønne Prisle, Jack Lin, Jussi Malila
4.7.2019
University of OuluDepartment of Physics
ATMOSMaster’s thesis
763683S
Abstract
The Saturnian moon Enceladus contains a liquid water ocean be-neath its surface. Ocean water is released into space through cracksin the moon’s frozen crust, forming a plume of vapor and ice-grains.NASA’s Cassini mission detected complex organic molecules in theseplume particles. The aim of this work is to numerically estimate thechemical composition of the ice-grains found in Enceladus’ plume.The ice-grains are assumed to form as bubble bursting aerosols at theocean surface. Known scaling laws of film and jet drop sizes are com-bined with a monolayer model of the liquid surface layer, a methodwhich could in principle be applied to any system with bubble burst-ing aerosols. The bulk ocean water is modeled as an aqueous solu-tion of sodium chloride, sodium carbonate, sodium bicarbonate andslightly soluble organic compounds. The following organic compoundsare considered as proxies for the organic compounds present on Ence-ladus: phenylalanine and its sodium salt as a proxy for an aromaticcompound and an amino acid, benzoic acid and benzyl alcohol. TheCassini measurements also imply the existence of very large organicmolecules on Enceladus, with molecular masses in excess of 200 u. Asa proxy for such compounds we have chosen the humic-like substanceSuwannee River Fulvic Acid. Both saturated and supersaturated casesfor the organic concentrations are considered. The calculations de-scribe highly enriched organic concentrations for smaller droplets andnearly bulk concentrations for larger ones, as one would expect fromsurface active compounds. The calculated droplets have lower concen-trations of salts than organics, in contrast with Cassini measurements.The calculations could be improved by better estimations of solutionsurface tension, more sophisticated surface layer characterizations anda better understanding of the bulk ocean composition. Results are alsohindered by a poor understanding of the bursting bubble distribution.Future models could account for aerosol dynamics as the droplets riseto the moon’ surface.
Contents
1 Introduction 11.1 Enceladus and physical conditions . . . . . . . . . . . . . . . . 2
2 Theory and background 32.1 Capillary waves . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Viscous dissipation . . . . . . . . . . . . . . . . . . . . 72.2 Dynamics of bubble bursting . . . . . . . . . . . . . . . . . . . 8
2.2.1 Film drop production . . . . . . . . . . . . . . . . . . . 92.2.2 Jet drop production . . . . . . . . . . . . . . . . . . . . 14
2.3 The monolayer model . . . . . . . . . . . . . . . . . . . . . . . 17
3 Calculations and results 183.1 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 Relevant bubble sizes . . . . . . . . . . . . . . . . . . . 203.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Conclusions 28
A Appendix: Surface tensions, viscosities, densities and solu-bilities 29
References 32
Nomenclature
γ Damping coefficient
δ Minimum thickness of the pinching region
ε Efficiency of cell puncture
θc Bubble half-cap angle
λ Wavelength of a capillary wave
λ0 Characteristic size of a convection cell
µ Dynamic viscosity
ν Molecular volume
ξ Position of a liquid surface
ρ Density
σ Surface tension
φ Velocity potential
χ Mole fraction
ω Angular frequency
a Capillary length
Bo Bond number
d Film drop diameter
E Energy
f0 Appearance frequency of convection cells
g Gravitational acceleration
h Thickness of the film cap
hb Critical thickness of the film cap
k Wavenumber
L Amplitude of a capillary wave
l Length of the pinching region
ls Surface layer thickness
lµ Viscous-capillary length
M Molar mass
m Mass
Oh Ohnesorge number
Oh1 First critical Ohnesorge number
Oh2 Second critical Ohnesorge number
P Perimeter of the film cap
p Pressure
R Bubble cavity radius
r Bubble cap radius
Rd Jet drop radius
RH Hinze scale
RBo Critical radius corresponding to Bo = 0.1
ROh1 Critical radius corresponding to Oh1
S Surface area of the film cap
t, τ Time
V Volume
v, u Velocity
vµ Viscous-capillary speed
w Mass fraction
x, y, z Cartesian coordinates
1 Introduction
Saturn’s geologically active moon Enceladus contains a liquid water oceanbeneath its icy surface [1]. Tidal heating of the porous core sustains theliquid state of the subsurface ocean [2]. Through cracks in the moon’s crust,ocean water is released into space [1]. This released material is responsi-ble for the formation of Saturn’s E ring [2]. Measurements made by massspectrometers aboard the Cassini spacecraft show that frozen ice-grains inthe plume contain complex organic compounds [2], making Enceladus a veryinteresting object of study. The origin of these organic compounds, whetherprimordial or biogenic, is unclear at present. In the primordial scenario theorganic carbon on Enceladus would predate the formation of the moon [2].Over time hydrothermal processes in the core would produce complex organiccompounds from simpler precursor molecules [2].
Our aim is to produce numerical estimates for the chemical compositionof these plume particles. We suppose that the ice-grains originate as bubblebursting aerosols at the ocean surface, which is a well studied phenomenonon Earth’s oceans [3]. On Earth, the bubbles are mostly a result of breakingwaves [3]. On Enceladus bubbles of entrapped volatile gases rise through theocean and burst at the surface [2], resulting in a spray of film drops and theejection of a drop from a central jet following cavity collapse. We combineknown scaling laws of film and jet drop sizes [4, 5] with a characterizationof the surface layer [6] to produce estimates of the chemical composition ofbubble bursting aerosols on Enceladus.
1
Figure 1: Image taken by the Cassini spacecraft. Water escapes into spacethrough cracks in Enceladus’ surface. Credits: NASA/JPL/Space ScienceInstitute
1.1 Enceladus and physical conditions
Enceladus’ ocean water is modelled as an aqueous solution of NaCl, NaHCO3,Na2CO3 and slightly soluble organic compounds [1]. The concentration ofNaCl is 0.05− 0.2 molal [1]. Na2CO3 and NaHCO3 are found in equal mea-sure, with a combined concentration 0.01 − 0.1 molal [1]. The pH of theocean is estimated to be ∼ 11-12 [1]. The following organic compounds areconsidered as proxies for the organic compounds present on Enceladus [2]:phenylalanine and its sodium salt as a proxy for an aromatic compound andan amino acid, benzoic acid and benzyl alcohol. The Cassini measurementsalso imply the existence of very large organic molecules on Enceladus, withmolecular masses in excess of 200 amomic mass units (u) [2]. As a proxy forsuch compounds we have chosen the humic-like substance Suwannee RiverFulvic Acid (SRFA).
Under such high pH conditions phenylalanine, with pKa = 9.76 [7], wouldbe significantly dissociated. At pH = 11 one would expect there to be threetimes as much sodium phenylalanine than phenylalanine present, and fivetimes as much at pH = 12 [7].
2
Like Glein et al. in their calculations, we shall consider the temperatureat the ocean surface to be at the triple point 0.01 ◦C, since the liquid oceanand the frozen crust should be in equilibrium [1].
To complete our model of ocean bulk composition we must choose theconcentrations of the considered organic compounds. Our first system willbe one with saturation concentrations at 0 ◦C. One possibility for the originof the organic compounds are chemical processes in the moon’s core, wherethe temperature would be much higher [2]. Our second system will be a su-persaturated case, with organic concentrations corresponding to solubilitiesat 100 ◦C, assuming that that is the temperature of the ocean-core interface[2].
The gravitational acceleration on Enceladus is 0.113 m/s2, roughly a per-cent of that on Earth [8].
2 Theory and background
Two mechanisms for plume particle formation are investigated: film dropsand jet drops, both produced by bubble bursting at the ocean-gas interface,analogous to sea spray aerosol formation on Earth [3]. Analysis of film dropformation is given by Lhuissier and Villermaux [4]. Scaling laws of jet dropsare given by Ganan-Calvo [5].
Ocean water is modeled as a pseudobinary mixture of an inorganic solu-tion and an organic component, and the surface composition is given by amonolayer partitioning model [6].
2.1 Capillary waves
In this section we will review the theory of capillary waves, following thetreatment of Landau and Lifshitz [9]. The dispersion and dissipation of cap-illary waves will be needed in section 2.2.2 when we consider the formationof jets from bursting bubbles.
Let ξ(x, y) denote the z-coordinate of a liquid surface. At rest the surfaceis the xy-plane at z = 0. If the liquid is perturbed from its equilibrium po-sition, a wave will propagate along the surface under the influence of gravity
3
and surface tension. Assume that the amplitude L of the wave is small com-pared to the wavelength λ. Let τ be the oscillation period. During this timethe fluid elements will move distances on the order of L, so their velocity willbe v ∼ L/τ . Fluid elements near the surface will experience acceleration
Dv
Dt=∂v
∂t+ (v · ∇)v. (1)
The velocity of fluid elements in a wave varies over the oscillation period τand the wavelength λ. Therefore, for the time and spatial derivatives
∂v
∂t∼ v
τ∼ L
τ 2, (2)
∇v ∼ v
λ∼ L
τλ. (3)
Comparing the two terms, one has
(v · ∇)v
∂tv∼ v2
λ
τ
v∼ vτ
λ∼ L
λ. (4)
Thus, the advective term (v · ∇)v can be neglected if L� λ, which is whatwe have assumed. The Euler equation is then
ρ∂v
∂t= ∇(−p− ρgz). (5)
Taking the curl of both sides, results in
ρ∇× ∂v
∂t= ρ
∂∇× v∂t
= 0, (6)
since the curl of the gradient is identically zero. This implies that the vorticity∇ × v is constant. However, in oscillatory motion, the time average of thevelocity should be zero
〈v〉 =1
τ
∫ τ
0
vdt = 0. (7)
Take again the curl of the above equation,
∇× 〈v〉 =1
τ
∫ τ
0
∇× vdt = 0. (8)
Since the vorticity is constant and the above integral is zero, the vorticitymust be zero.
4
For any irrotational flow, there exists a velocity potential φ from whichthe velocity can be derived,
v = ∇φ. (9)
Such flows are called potential flows. If a potential flow is incompressiblesuch that ∇·v = 0, it immediately follows that the velocity potential satisfiesLaplace’s equation
∆φ = 0. (10)
We have seen that waves of small amplitudes can be approximated aspotential flows. Thus we can write Euler’s equation as
∇(ρ∂φ
∂t+ p+ ρgξ) = 0 (11)
This implies that the bracketed term is some function f(t) of time only, butthis function can be absorbed into φ by writing φ′ = φ +
∫f(t)dt. This has
no physical significance since both φ and φ′ will yield the same velocity field.Therefore we can set f(t) = 0 and write
p = −ρgξ − ρ∂φ∂t. (12)
The pressure in a fluid near the surface is given by the Young-Laplace equa-tion
p− p0 = σ
(1
R1
+1
R2
), (13)
where σ is the surface tension of the fluid, R1 and R2 the principal radii ofcurvature and p0 is some constant pressure on the surface, for example theatmosphere. We have assumed that the amplitude of the wave is small, soξ(x, y) is small as well. For such slightly curved surfaces the bracketed termin equation (13) can be approximated as
1
R1
+1
R2
= −(∂2ξ
∂x2+∂2ξ
∂y2
). (14)
Combining equations (12), (13) and (14), we have that
ρ∂φ
∂t+ ρgξ − σ
(∂2ξ
∂x2+∂2ξ
∂y2
)= 0, (15)
5
where we have gotten rid of p0 by again redefining φ as φ − p0t/ρ. As anapproximation, suppose that z-component of velocity near the surface is justthe time derivative of ξ,
vz =∂φ
∂z=∂ξ
∂t. (16)
Take the time derivative of (15),
ρ∂2φ
∂t2+ ρg
∂ξ
∂t− σ ∂
∂t
(∂2ξ
∂x2+∂2ξ
∂y2
)= 0
⇒ρ∂2φ
∂t2+ ρg
∂ξ
∂t− σ
(∂2
∂x2∂ξ
∂t+
∂2
∂y2∂ξ
∂t
)= 0
⇒ρ∂2φ
∂t2+ ρg
∂φ
∂z− σ
(∂2
∂x2∂φ
∂z+
∂2
∂y2∂φ
∂z
)= 0, z = 0.
(17)
The potential φ must now satisfy Laplace’s equation (10) with the boundarycondition (17). Consider a plane wave propagating along the x-axis. Solu-tions are of the form φ = f(z) cos (kx− ωt), where k is the wave number andω the angular frequency.
∂2φ
∂x2+∂2φ
∂z2= 0
⇒ −k2 cos (kx− ωt) +∂2f(z)
∂z2cos (kx− ωt) = 0
⇒ ∂2f(z)
∂z2= k2f(z).
(18)
The solution is either f(z) = Aekz or Ae−kz. We choose f(z) = Aekz,since the velocity potential must diminish inside the fluid. The disper-sion relation is obtained from the boundary condition (17) by insertingφ = Aekz cos (kx− ωt),
ρ∂2φ
∂t2+ ρg
∂φ
∂z− σ
(∂2
∂x2∂φ
∂z+
∂2
∂y2∂φ
∂z
)= 0
⇒ ρgk − ρω2 − σ(−k3 + 0) = 0
⇒ ω2 = gk +σ
ρk3.
(19)
Comparing the two terms on the right in equation (19), one obtains thedimensionless Bond number Bo = ρgλ2/σ for the wave,
gk
k3σ/ρ=
gρ
k2σ=ρgλ2
σ, (20)
6
where the wavelength λ is taken to be the characteristic length of the system.Thus the Bond number compares gravitational and surface tension forces. IfBo� 1, gravitational effects can be neglected and the wave is called a cap-illary wave. If Bo � 1, the surface tension term is negligible and we havegravity waves. Intermediate cases, Bo ≈ 1, are called capillary gravity waves.
For capillary waves the dispersion relation becomes ω2 = σk3/ρ, and thepropagation speed of the wave is
∂ω
∂k=
3
2
(σk
ρ
)1/2
. (21)
2.1.1 Viscous dissipation
The energy dissipation due to internal friction in an incompressible flow is
E = −1
2µ
∫ (∂vi∂xk
+∂vk∂xi
)2
dV, (22)
where µ is the dynamic viscosity of the fluid. Note that we are here assumingthat the viscosity is uniform. This is not necessarily true in the presence ofsurfactants, which may introduce a vertical viscosity gradient. Now, for apotential flow ∂vi/∂xk = ∂2φ/∂xk∂xi = ∂vk/∂xi, so
E = −1
2µ
∫ (2∂vi∂xk
)2
dV = −2µ
∫ (∂2φ
∂xi∂xk
)2
dV
= −2µ
∫(−k2 sin (kx− ωt)Aekz)2dV
= −2µk4∫A2e2kz sin2 (kx− ωt)dV,
(23)
the time average of which is
〈E〉 = −2µk4∫〈φ2〉 dV, (24)
since〈φ2〉 = A2e2kz 〈cos2(kx− ωt)〉 , (25)
and 〈sin2(kx− ωt)〉 = 〈cos2(kx− ωt)〉 . For any periodic motion with a smallamplitude, the average kinetic and potential energies are equal, so the mean
7
total mechanical energy of the system can be written as twice the meankinetic energy
〈E〉 = ρ
∫〈v2〉 dV = ρ
∫〈(∇φ)2〉 dV
= ρ
∫k2 〈sin2(kx− ωt)〉A2e2kz + k2 〈cos2(kx− ωt)〉A2e2kzdV
= 2ρk2∫〈φ2〉 dV.
(26)
We define the damping coefficient γ as the ratio of the energy dissipationand total energy
γ =|〈E〉|〈E〉
=µk2
ρ, (27)
which has units of time−1. We may define another dimensionless numberby multiplying the damping coefficient by a characteristic timescale τ of thesystem. A natural choice in this case is the time it takes for the wave to prop-agate the distance one wavelength. For a capillary wave with a propagationspeed v ∼ (σk/ρ)1/2, we have
γτ =γλ
v=
µ
ρλ2λ3/2ρ1/2
σ1/2=
µ√ρσλ
. (28)
Equation (28) defines the Ohensorge number Oh = µ/√ρσλ of the system,
with the wavelength λ being here the characteristic length of the system.The motion of the wave is driven by surface tension forces and is dissipatedby viscosity. The Ohnesorge number compares the viscous forces with thecapillary forces, and if Oh � 1, the wave will be damped very rapidly. IfOh ≈ 1, then the wave will be damped over a distance of one wavelength,and if Oh� 1 then the wave may propagate distances much larger than onewavelength before being dissipated.
2.2 Dynamics of bubble bursting
Bubble bursting at a liquid surface produces two kinds of aerosols. After abubble bursts, the rippling film cap breaks up into a large number of dropletscalled film drops [4]. Capillary waves then travel down the exposed bubblecavity, colliding in the centre and producing a rising jet of liquid, which maybreak up into jet drops [5]. Jet drops are generally less numerous than filmsdrops [3].
8
Bubble bursting is an important physical process, as the resulting aerosolsprovide a mechanism of material and heat exchange between the ocean andthe atmosphere [3]. On Earth the droplets regulate atmospheric chemistry,global radiation balance and the water cycle [10].
The chemical composition of smaller droplets may differ significantly fromthe bulk composition if surface-active compounds are present [3], as is thecase on Enceladus. In this section we will review the theory of bubble burst-ing and the resulting droplet production, with the most important resultsbeing the formulae of droplet radii as functions of the bursting bubble ra-dius [4][5]. In section 3 we will combine this knowledge with the monolayermodel [6] reviewed in section 2.3 to produce estimations of the compositionof Enceladus’ plume particles.
2.2.1 Film drop production
The analysis of film drop production is given by Lhuissier and Villermaux intheir 2011 paper [4]. In this section we will review the steps taken to arriveat the scaling law for average drop size.
Figure 2: Diagram of a bubble at a liquid interface
θc
r
Consider first a submerged bubble at the liquid surface. The bubbleprofile will consist of three regions; the cavity surface with radius of curvatureR, the bubble cap with radius of curvature r, and the meniscus where thecap connects to the liquid bulk, see figure 2. The cavity radius is required tobe half the cap radius, due to the cap having two interfaces instead of one,like the cavity surface. We will assume that the density of the entrappedgas ρ0 is negligible compared to that of the surrounding liquid ρ, such that
9
ρ− ρo ≈ ρ. Then due to gravity, the bubble experiences a buoyancy
ρg4π
3
(r2
)2, (29)
which is opposed by the vertical component of surface tension acting on therim of the bubble cap
2πrσ sin2 θc. (30)
The bubble comes to rest when the two forces are equal. From this one canobtain for the half-cap angle θc
sin θc =
√ρg
σ
r
2√
3=
r
2√
3a=
√Bo
2√
3, (31)
where a =√σ/ρg is the capillary length and the Bond number Bo =
ρgr2/σ = (r/a)2 again compares the gravitational and surface tension forces.If Bo is sufficiently small, one can approximate θc as
θc =
√Bo
2√
3, (32)
which holds up to Bo = 25 [4]. Since gravity and surface tension are the onlyforces acting on the bubble, it is not surprising that the bubble geometry isdefined by the Bond number. In the limit of Bo → 0 the bubble becomescompletely submerged and spherical, resting just below the surface. In sucha case one would not expect any film drops to form, since no film cap exists.Therefore there should exist some critical Bond number below which no filmdrops are produced.
Once the bubble is at rest, gravity causes a surfactant concentration gra-dient and thus a surface tension gradient to form, holding up the film cap[4]. The pressure inside the cap is the sum of capillary pressure, due tothe Young-Laplace equation (13), and hydrostatic pressure. The capillarypressure will dominate in the case of Bo < 25 [4]. The pressure differencebetween the meniscus and the film interior is the capillary pressure 2σ/ρ,which drives the draining flow u, from the film to the meniscus. This flowcreates a pinching region at the perimeter of the film cap, with length l andminimum thickness δ [4], see figure 3. The flow is in the interior of the film,between two mobile layers of surfactants. For a steady flow, one has
ρDu
Dt= −∇p+ µ∇2u = 0. (33)
10
Figure 3: Diagram of the pinching region
hδ
l
The pressure difference 2σ/r is over the length l. Assuming no-slip at thefilm interfaces, the velocity changes over the distance δ. Then in orders ofmagnitude
σ
rl∼ µu
δ2. (34)
The curvature in the pinching region should match the cap curvature [4]
h− δl2∼ 1
r. (35)
At the same time an opposite flux exists in the surface layer. Convectivemotion is induced over the pinching region due to destabilization caused bythe surface tension difference [4]. This phenomenon is called marginal regen-eration, and it regulates the pinching region such that the neck thickness δnever becomes smaller than half the film cap thickness h.
Now, from (34) we have that
u ∼ σδ2
rlµ, (36)
and from (35)
l ∼√r(h− δ) ∼
√rh. (37)
11
Therefore the drainage velocity
u ∼ σδ2
r√rhµ
=σh2
µr3/2h1/2=σ
µ
(h
r
)3/2
. (38)
The liquid drains from the film cap with thickness h and surface area S,through the cap perimeter P . From mass conservation, one has for theinterior flow
h ∼ −huPS. (39)
The surface area of the cap
S = 2πr2(1− cos θc) ≈ πr2θ2c , (40)
and the perimeterP = 2πr sin θc ≈ 2πrθc, (41)
where the small-angle approximations have been used. The ratio P/S is then
P
S∼ 2πrθcπr2θ2c
=2
rθc. (42)
From (32), (38), (39) and (42), we have that
h ∼ −hu 2
rθc∼ −σah
5/2
µr7/2, (43)
which is a separable differential equation. Separating and integrating (43),we get a thinning law for the film thickness
h ∼(µr7/2
σat
)2/3
. (44)
Convection cells from marginal regeneration appear at a frequency [4]
f0 ∼σ
µa
(r
a
)1/3 (µaσt
)2/3, (45)
and have a characteristic size [4]
λ0 ∼ r
(h
r
)3/2
. (46)
12
It has been observed that film puncture happens near the cap foot, in thecentre of these convection cells [4]. However this process is seen to be ineffi-cient; Lhuissier and Villermaux propose ε ∼ O(10−4− 10−3) as the efficiencyof cell puncture, such that the probability of puncture in a cell within a timeinterval δt is εf0δt. The probability p(t) of a puncture somewhere on thesurface is then proportional to ε and the number of cells, which is ∼ P/λ0,
p(t)δt ∼ εP
λ0f0δt. (47)
Using equations (41), (45), (46) and defining
τ0 =(4/3)3/4
ε3/4µa
σ
(r
a
)1/2
, (48)
we write the probability p(t) as
p(t) =4
3
t1/3
τ4/30
. (49)
If the bubble has not burst from time t = 0 to some time t, it must nothave burst on subsequent time intervals δt′ from 0 to t. The probability ofno puncture is
Q(t) =
t/δt′∏t′/δt′=0
[1− p(t′)δt′], (50)
which becomes
Q(t) = exp(−∫ t
0
p(t′)dt′)
(51)
and in the limit of δt→ 0. The mean bubble lifetime is [4]
tb =
∫ ∞0
Q(t)dt = Γ(7/4)τ0 ≈ 0.92τ0. (52)
With the thinning law (44) and the above mean lifetime, one has the filmthickness at time of bursting
hb ∼(µr7/2
σaτ0
)2/3
= aε1/2(r
a
)2
= aε1/2Bo. (53)
After puncture the nucleated hole is expanded by surface tension with avelocity v =
√2σ/ρh [4]. The rim recedes following the film curvature and
13
experiences a destabilization which causes ligaments to form at the edge ofthe rim. Ligaments are spaced by the instability wavelength λ ∼
√rh [4]
and they form with a growth time τ ∼√ρ(rh)3/2/σ [4]. The ligaments then
break up into film drops, and their diameter is set by the ligament diameterat breakup. The ligament diameter is seen to remain constant from themoment of formation τ , since their stretching is compensated by flow fromthe film [4]. The average ligament and thus drop diameter is proportional tothe rim diameter at the time of ligament formation τ [4]
〈d〉 ∼√vτhb ∼ r3/8h
5/8b . (54)
As a crude approximation, the number of film drops produced can beestimated by assuming the entire volume of the film cap ∼ Shb is formedinto film drops with volume ∼ 〈d〉3,
〈N〉 ∼ Shb
〈d〉3∼ r2hbBo
r9/8h15/8b
= Bo
(r
hb
)7/8
. (55)
For film drops to form, the recession time of the rim ∼ rθc/V must belarger than the growth time τ and the ejection time ∼ 3τ [4]. In the limitingcase √
ρ(rhb)3/2
σ∼ rθc/v ∼ r
√Bo/
√2σ/ρhb
Bo ∼ ρr3/2h3/2b σ
σr2ρhb=
(hbr
)1/2
Bo ∼ (Boε)1/4
Bo ∼ ε1/3,
(56)
and since ε ∼ 10−3, we have the Bond number below which no film dropsshould form
Bo ∼ O(10−1). (57)
2.2.2 Jet drop production
Once the bubble cap has burst, capillary waves will travel down the exposedcavity, colliding at the axis of symmetry and forming an upwards liquid jet.
14
A scaling law for the ejected jet drop is given by Ganan-Calvo [5]. Thecollapsing capillary waves have a propagation speed v0 given by the equation(21), and experience viscous dissipation from the equation (27) as they traveldown the cavity. Assuming that the wavelength λ of the capillary waves isproportional to the cavity radius R [5], we have that
λ = 1/k ∼ R
v0 ∼(σ
ρR
)1/2 (58)
The length travelled by the capillary wave is proportional to R, so its prop-
agation rate is τ−1c ∼ v0/R ∼(
σρR3
)1/2
. The dissipation rate from equation
(27) is γ ∼ µρR2 . In order to form a jet and eject a drop, the wave must not
be completely dissipated before it reaches the bottom of the cavity. In otherwords, the propagation rate must be larger than the dissipation rate
τ−1c > γ
τcγ < 1.(59)
We have the Ohnesorge number from equation (28)
Oh = γτc =µ√ρσλ
< 1. (60)
The Ohnesorge number must be at the very least less than one. However, atthe time of collapse the wave must still have significant energy left to ejecta drop, and the actual critical Ohnesorge number has been experimentallydetermined to be Oh1 = 0.038 [5]. The Ohnesorge number of the systemmust be lower than Oh1 to eject a jet drop.
Figure 4: Diagram of jet formation
Rd
L
u
vj
15
Let us identify characteristic speeds and length scales at the moment ofcollapse. Let the eventual drop radius Rd be the radial length and the ampli-tude of the wave L the axial length. Let u be the radial speed in the surfacelayer and the launch speed of the jet vj the axial speed. See figure 4.
Now, the drop is seen to be ejected at a height comparable to R, sothe jet requires a kinetic energy ∼ ρRR2
dv2j to eject a drop [5]. The energy
available to the jet is the surface energy of the ruptured film cap ∼ σS,the gravity potential imbalance of the cavity proportional to its height andvolume ∼ (ρgR)R3, and the negative contribution of the viscous dissipation∼ µ(σR3/ρ)1/2. Thus we have the energy balance
Oh1σR2 + kBo,1ρgR
4 − µ(σR3/ρ)1/2 ∼ k′ρRR2dv
2j , (61)
where kBo,1 and k′ are fitting parameters [5]. There is also the negative con-tribution from gravity acting on the rising jet, but this is negligible, sincethe volume of the jet is much smaller than that of the cavity.
For the momentum balance we have the observation that maximum ve-locity takes place just after collapse, as the jet is about to form [5]. Thereforeat the moment of jet ejection we have near zero stress for a fluid element atthe surface near the axis
ρ∂v
∂t+ ρ(v · ∇)v = −∇p+ µ∇2v ∼ 0, (62)
and thus all four terms should be comparable to each other. In orders ofmagnitude
ρv2j/L ∼ σ/R2d ∼ µu/L2. (63)
Considering a cylindrical control volume at the point of collapse, with radiusRd and height L. One has from mass conservation
uLRd ∼ vjR2d, (64)
since we are assuming an incompressible flow. From equations (63) and (64)
Rd/lµ ∼ (u/vµ)−5/3,
L/lµ ∼ (u/vµ)−4/3,
vj/vµ ∼ (u/vµ)2/3,
(65)
where vµ = σ/µ and lµ = µ2/ρσ are the viscous-capillary speed and length,respectively [5].
16
The above scaling laws (65) should hold for any capillary wave collapsingat the cavity bottom, and it is seen that faster but smaller wave collide beforea slower but larger wave, with λ ∼ R, produces a jet. Ganan-Calvo proposesthat the radial speed u is induced by the capillary waves through viscouseffects and the inertial push of the largest wave [5]
ρu2 ∼ µvc/L+Oh2ρv20, (66)
where Oh2 is a second critical Ohnesorge number signifying the inviscid limit,below which the inertial push of the largest wave will dominate. The valueof Oh2, is experimentally determined to be Oh2 = 0.0045 [5].
Now, combining equations (61), (65) and (66), one has the scaling lawfor the ejected drop radius [5]
Rd
lµ∼
[Oh−1(Oh1Oh− 1 + kBo,1
BoOh
)]5/4(
1 + Oh2Oh
)1/2 ≡ φR, (67)
Rd
lµ= kdφR, (68)
where the fitting parameters kBo,1 = 0.006 and kd = 0.9 [5].
2.3 The monolayer model
In order to compute the chemical composition of the produced aerosols, onehas to know the composition and thickness of the surface layer. The tools toestimate these are provided by Malila and Prisle [6].
Let χb = (χb1, χb2, ..., χ
bi) be the composition of the liquid bulk, where χbi
are the mole fractions of the constituent compounds of the solution. Let χs
be the corresponding surface layer composition. Let σi be the surface tensionof the pure compounds i that make up the solution, and νi their molecularvolumes. The surface tension of a droplet, with surface layer of thickness ls,is given by the following semiempirical formula [6]
σ(χb) =Σiσiνiχ
si
Σiνiχsi, (69)
and the surface layer thickness by [6]
ls =
(6
π
∑i
νiχsi
)1/3
. (70)
17
The molecular volumes of the components are calculated from their molarmasses Mi and densities ρi as
νi =Mi
ρiNA
, (71)
where NA is the Avogadro constant.
The solution of surface composition χs from equation (69) using pseu-dobinary approximations will be covered in section 3.
3 Calculations and results
3.1 Calculations
Using the tools reviewed in the last section, we can estimate the chemicalcomposition Enceladus’ plume particles in terms of mass fractions. The cal-culations are carried out in MATLAB.
We take as input the liquid bulk composition χb and the radius of thebursting bubble. To start out, we can estimate the surface tension σm andviscosity µm of a mixture using the following mixing formulas [11], for thesurface tension
lnσm =∑i
χbi lnσi, (72)
and the viscosity
µm =∏i
µχbii . (73)
See Appendix A for a complete description of the component and mixturesurface tensions, viscosities and densities.
From the bubble radius R, by which we here mean the cavity radius, wecan calculate the capillary length a =
√σ/ρg, the viscous-capillary length
lµ = µ2/ρσ, the Ohnesorge number Oh = µ/√ρσR and the Bond number
Bo = ρgR2/σ. With these values the radius r of the produced droplets iscalculated either from the equation (54) or the equation (68), depending onthe Bond number. For film drop formation it is required that Bo > 0.1, andsince the number of film drops generated is in general much larger than thenumber of jet drops, we will neglect jet drop formation in this region. Thevolume of the droplet is of course
Vd =4
3πr3, (74)
18
and with the thickness of the surface layer from equation (70), the volumeof the surface layer is
Vs = Vd − Vb, (75)
where the droplet bulk volume Vb is
Vb =4
3π(r − ls)3. (76)
Now we must acquire the surface layer composition χs from equation(69). We are considering a system with water and four components, andwe cannot simultanously solve five numbers from one equation. For twocomponents however, the equation can be solved as
χ2 =ν1σ1 − σν1
σ(ν2 − ν1) + ν1σ1 − ν2σ2, (77)
since χ1 = 1 − χ2. We will use a pseudobinary approximation, where weconsider the system to be a mixture of only two components [6], the organicpart χorg and salt water χsw = χw + χ2 + χ3 + χ4. The molecular volume ofthe pseudobinary salt water is approximated as [6]
νsw =
∑i χ
biMi
ρsw, (78)
where χbi = χbi/Σjχbj are the pseudobinary salt water mole fractions, and the
sums are over the components of the pseudobinary mixture, including water.The salt water density ρsw is again given in appendix A.
The surface mole fraction for the organic compound is then
χorg =νswσsw − σmνsw
σm(νorg − νsw) + νswσsw − νorgσorg, (79)
where σm is the surface tension of the mixture and σsw the surface tensionof salt water. The rest of the mole fractions are solved one at a time usingfurther pseudobinary approximations, taking one of the salts on its own andhaving the rest make up a new pseudobinary mixture.
Having calculated the droplet bulk and surface volumes Vb and Vs, andthe surface composition χs, we get the mass of the organic compound in thesurface layer
msorg = wsorgms, (80)
19
where wsorg = χsorgMorg/ΣiχsiMi is the mass fraction of the organic compound
in the surface layer, and ms = Vsρs is the total mass of the surface layer.Similarly for the droplet bulk, we have
mborg = wborgmb, (81)
with wborg = χborgMorg/ΣiχbiMi and mb = Vbρb. The mass fraction of the
organic compound in the entire droplet is then
worg =msorg +mb
org
md
, (82)
where md = ms +mb is the total mass of the droplet.
3.1.1 Relevant bubble sizes
Assuming that the viscosity, surface tension and gravitational accelerationcan be considered as being roughly constant during the bubble bursting pro-cess, the Bond and Ohnesorge numbers become essentially functions of thebubble radius. Then, the critical Bond and Ohnesorge numbers from section2.2 already place some constrains upon the considered bubble size spectrum.The lower bound ROh1 for the size of a bubble is placed by the critical Ohne-sorge number Oh1 = 0.038. The upper bound for jet drop formation is givenby Bo ≈ 3 [12], but we already decided to neglect jet drop formation aboveBo ≈ 0.1, where the film drop mechanism dominates. Let RBo be the criticalradius corresponding to Bo = 0.1.
We will consider the upper bound of bubble distribution to be given bythe Hinze scale [13]
RH = c(σ/ρ)3/5γ−2/5t , (83)
where c is a constant and γt is the turbulent dissipation rate experiencedby a bubble moving through the liquid. A bubble with larger radius thanthe Hinze scale is likely to break up into smaller bubbles under turbulence[13]. We will use the estimated values c = 0.363 and γt = 0.1 W/kg [13] tocalculate the Hinze scale.
3.2 Results
The critical radii introduced in section 3.1.1 are listed in table 3.2 for eachorganic compound. We see that in each case the Hinze scale RH , which as theupper bound for our bubble radius, is less than the critical radius RBo. Thuswe are always in the range Bo < 0.1 and considering jet drop production
20
only. In figures 5-9 we have calculated the organic mass percentages fromequation (82) in the range ROh1 −RH .
Table 1: The critical radii in meters for each organic compound, supersatu-rated cases in brackets
Compound RH ROh1 RBo
Benzyl alcohol 1.8 · 10−3 8.3 · 10−5 6.7 · 10−3
Benzoic acid 3.1 · 10−3 3.1 · 10−5 1.0 · 10−2
SRFA 2.3 · 10−3 4.9 · 10−5 8.1 · 10−3
Phenylalanine 3.1 · 10−3 3.3 · 10−5 1.0 · 10−2
Sodium phenylalanine 3.1 · 10−3 4.0 · 10−5 1.0 · 10−2
(Benzyl alcohol) 6.4 · 10−4 5.0 · 10−4 2.8 · 10−3
(Benzoic acid) 3.1 · 10−3 3.3 · 10−5 1.0 · 10−2
(Phenylalanine) 3.0 · 10−3 4.4 · 10−5 1.0 · 10−2
(Sodium phenylalanine) 2.9 · 10−3 9.3 · 10−5 1.0 · 10−2
10-4
10-3
Radius (m)
0
0.2
0.4
0.6
0.8
1
Ma
ss p
erc
en
tag
e
Low solubility
High Solubility
Figure 5: Mass percentage of SRFA in jet drops as a function of burstingbubble radius. Corresponding jet drop radius range (m): 5.2·10−10−3.7·10−4
(supersaturated: 1.2 · 10−9 − 2.7 · 10−4)
21
10-4
10-3
Radius (m)
0
0.2
0.4
0.6
0.8
1
Mass p
erc
enta
ge
Saturated
Supersaturated
Figure 6: Mass percentage of benzyl alcohol in jet drops as a function ofbursting bubble radius. Corresponding jet drop radius range (m): 7.3·10−10−1.8 · 10−4 (supersaturated: 5.0 · 10−9 − 3.4 · 10−6)
10-5
10-4
10-3
10-2
Radius (m)
0
0.005
0.01
0.015
0.02
0.025
Mass p
erc
enta
ge
Saturated
Supersaturated
Figure 7: Mass percentage of benzoic acid in jet drops as a function ofbursting bubble radius. Corresponding jet drop radius range (m): 6.0·10−10−4.5 · 10−4 (supersaturated: 5.8 · 10−10 − 4.4 · 10−4)
22
10-4
10-3
Radius (m)
0
0.05
0.1
0.15
0.2
0.25
Mass p
erc
enta
ge
Saturated
Supersaturated
Figure 8: Mass percentage of phenylalanine in jet drops as a function ofbursting bubble radius. Corresponding jet drop radius range (m): 4.4·10−10−4.4 · 10−4 (supersaturated: 4.2 · 10−10 − 4.1 · 10−4)
10-5
10-4
10-3
10-2
Radius (m)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Mass p
erc
enta
ge
Saturated
Supersaturated
Figure 9: Mass percentage of sodium phenylalanine in jet drops as a functionof bursting bubble radius. Corresponding jet drop radius range (m): 6.2 ·10−10 − 4.2 · 10−4 (supersaturated: 4.4 · 10−10 − 3.3 · 10−4)
23
In each curve we can identify two regions. In the immediate vicinity of thecritical radius ROh1 , the organic is enriched compared to the bulk concentra-tion. This is because the jet drop radius becomes comparable to the surfacelayer thickness, and thus the composition of the entire drop approaches thesurface layer composition. This is most clearly seen in figure 6. In thesecases, when the droplet is composed entirely or mostly of the surface layer,the accuracy of our calculation is determined by the accuracy of the chosensurface layer model, the monolayer model in this work.
The surface mole fraction of benzyl alcohol, as calculated from the mono-layer model, is equal to one in both the saturated and supersaturated cases.In the saturated case the drop radius becomes equal to the surface thicknessand we have a droplet composed entirely of benzyl alcohol (similar behavioris seen in the high solubility case for SRFA.) In the supersaturated case thesurface layer is still entirely benzyl alcohol, but the surface layer thicknesshas become significantly smaller than the drop radius and we have a masspercentage of roughly 40%.
As the bursting bubble radius increases, the drop composition very quicklybecomes nearly equal to the bulk composition, as the surface layer thicknessbecomes insignificant compared to the size of the jet drop. In light of thisbehavior, any uncertainties in the value of the Hinze scale due to γt becomeirrelevant, as long as RH � ROh1 .
In figures 10-13 we have calculated four systems, the saturated and super-saturated cases according to table 3, at high and low bubble radii, showingthe mass percentage of each component in the system.
24
0 50 100 150 200 250
Molecular mass (u)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Mass p
erc
enta
ge
NaCl Benzoic acid
Phenylalanine
Sodium phenylalanine
NaHCO3
Na2CO
3
Figure 10: Saturated system, calculated at the critical radius ROh1
0 50 100 150 200 250
Molecular mass (u)
0
0.1
0.2
0.3
0.4
0.5
Mass p
erc
enta
ge
NaCl
Benzoic acid
Benzyl alcohol
Phenylalanine
Sodium phenylalanine
Na2CO
3
NaHCO3
Figure 11: Supersaturated system, calculated at the critical radius ROh1
25
0 50 100 150 200 250
Molecular mass (u)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Mass p
erc
enta
ge
Benzoic acid
Phenylalanine
NaCl
Benzyl alcohol
Sodium phenylalanine
Na2CO
3
NaHCO3
Figure 12: Saturated system, calculated at the Hinze scale RH
0 50 100 150 200 250
Molecular mass (u)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Mass p
erc
enta
ge
NaCl
Benzyl alcohol
Benzoic acid
Sodium phenylalanine
Phenylalanine
Na2CO
3
NaHCO3
Figure 13: Supersaturated system, calculated at the Hinze scale RH
26
The latter two figures are calculated at the Hinze scale, but due to the be-havior seen in figures 5-9 these figures essentially describe droplets with bulkocean composition and could be produced from any bursting bubble witha radius suitably larger than the critical radius ROh1 . Indeed such dropletswould be produced by film drops as well, if bubbles larger than the Hinzescale are present. Figures 10 and 11 are calculated at the critical radius andshould be seen the upper limit for each compound. Benzyl alcohol, whosemass percentage goes to one in the saturated case, is omitted from figure10. The mass percentages of SRFA are 100% and 65% at the critical radius,and 1.9% and 0.2% at the Hinze scale, for the high and low solubility casesrespectively.
One should note that the organic compounds are treated separately anddo not interact with each other. The figures 10-13 are combined plots fromseparate calculations. The salt mass percentages do not vary greatly betweendifferent different organic compounds considered, and are layered on top ofeach other in the figures.
Our calculations describe droplets right after their formation. As thedroplets rise to the moon’s surface, their water content is liable to changedue to condensation from the surrounding vapor [2]. The total amount ofsalt and organics in the droplets however should stay the same. In theCassini measurements, the ratio of the benzene cation peak to the high-mass organic cation (HMOC) peaks is roughly 3:1 [2]. In our calculationsthis would correspond to the ratio of SRFA to the rest of the organic com-pounds. The system that best matches this ratio is the one seen in figure 12,with (wNaPhe +wPhe +wBnA +wBnOH)/wSRFA ≈ 3.3, if one chooses the highsolubility estimate for SRFA.
Comparing the sodium peaks to the organic peaks in [2], we see that thereis roughly twice as much salt in the ice grains than organic compounds. Noneof our calculated systems contain this much salt compared to the organics.The surface concentrations of the salts as calculated from the monolayermodel are roughly 60% of the bulk concentration. This behavior is consis-tent with experiments [14, 15]. If some of our calculated systems containedhigh amounts of salts, this discrepancy could be explained by a bubble sizespectrum that suitably emphasised those bubble sizes. However, the organicmass percentages are much higher than the salt mass percentages in everysystem, including the systems calculated at the Hinze scale. Thus, no bub-ble size spectrum could produce a system of ice grains with overall more saltthan organics. This implies that by assuming saturation concentrations in
27
the ocean we have introduced too much organic material into the systemfrom the beginning, and the bulk organic concentrations should perhaps belower than saturation. However, our model neglects any aerosol dynamicalprocesses the droplets might experience as they travel through the plume gasto the surface, which could explain the discrepancy.
4 Conclusions
By combining the scaling laws of film and jet drops with a surface monolayermodel, we have produced a method for estimating the chemical composi-tions of bubble bursting aerosols. We have applied this method to Ence-ladus’ plume ice grains which are known to contain organic compounds, butthe method has great generality and could be applied to any terrestrial orextraterrestrial system where bubble bursting aerosols might be found. Themodel predicts the strong enrichment of small droplets containing surfaceactive compounds. The calculated droplets contain lower amounts of saltsthan organic compounds, in contrast with Cassini measurements.
Characterization of the surface layer is essential to obtaining the com-position of a droplet. The calculations could be performed using a differentsurface layer model than the monolayer model used here. The calculationscould be improved by more sophisticated estimates for mixture surface ten-sions and viscosities, and a better understanding of the ocean water compo-sition. Results are also hindered by a poor knowledge of the bursting bubblesize distribution. Future models could account for aerosol dynamics as thedroplets travel from the ocean-gas interface to the moon’s surface.
28
A Appendix: Surface tensions, viscosities, den-
sities and solubilities
Water
The surface tension of water (N/m) is given by the following equation [16]
σw(T ) = 0.2358
(647.096− T
647.096
)1.256(1− 0.625
647.096− T647.096
), (84)
in the range 273 K < T < 380 K.
The viscosity of water (mPa s) is given by [17]
µw(t) =t+ 243
0.05594t2 + 5.2842t+ 137.37, (85)
in the range 0◦C < t < 150◦C.
The density of water (g/cm3) is given by [18]
ρw(T ) = 0.08 tanh
(T − 225
46.2
)+ 0.7415
(647.096− T
647.096
)0.33
+ 0.32. (86)
Sodium chloride
The extrapolated surface tension (N/m) of supercooled NaCl is given by[11]
σNaCl(T ) = 0.19116− 0.07188 · 10−3T. (87)
The surface tension of a binary water + salt mixture is given by [11]
σ = σw + χsFws(T ), (88)
where χs is the mole fraction of the salt and
Fws(T ) = a+ bT. (89)
The surface tension of a mixture of N components is then given by [11]
lnσ =N∑i
χi ln (σi(T ) +N∑j
χjFij(T )), (90)
29
which is a modified version of equation (72). See table 2 for the fittingparameters a and b for each compound.The extrapolated density (kg/m3) of liquid NaCl, based on, is given by [19]
ρNaCl = (2.1389− 0.5426 · 10−3T ). (91)
The relative viscosity µr = µs/µw of NaCl solution is given by Zhang andHan [20].
Sodium carbonate and sodium bicarbonate
The extrapolated surface tensions of supercooled sodium carbonate andbicarbonate, and their model parameters a and b are given by Dutcher et al.[11], see Table 2.
The densities of supercooled molten sodium carbonate and bicarbonateare extrapolated from the data given by Janz [19]. The densities of aqueoussodium carbonate and bicarbonate solutions are given by [21]. The densityof a salt water mixture with, sodium chloride, sodium carbonate and sodiumbicarbonate is given by the model of Potter and Haas [22].
The viscosity (mPa s) of a sodium carbonate and bicarbonate solution isgiven by the following fit
µ(t) = 1.8113− 0.0229t, (92)
made to values provided by [23].
Organic compoundsThe surface tension of pure benzyl alcohol is calculated from a fit given
by [24]. The surface tension of pure benzoic acid is extrapolated from thevalues given by [25].
The Dutcher model parameters are listed in the table below.
30
Table 2: Dutcher model parametersa (mN m−1) b (mN m−1K−1) Source
Benzyl alcohol -8824.6 8.7 [26]Benzoic acid -267.7 - [27]Phenylalanine 988.9 -9.4 [28]Sodium phenylalanine 693.8 -4.6 [29]NaCl 232.5 -0.25 [11]NaHCO3 46.4 - [11]Na2CO3 56.0 - [11]
The surface tensions of pure phenylalanine and sodium phenylalanine areset to 60 mN/m, as an estimate based on their aqueous solutions [28, 29]. ForSRFA, we replace the second term in equation (88) with the logarithmic termin the Szyszkowski equation [30]. The surface tension of pure SRFA is set to50 mN/m, again as a guess based on the aqueous solution surface tension [30].
The viscosity of benzyl alcohol at 0 ◦C is extrapolated from values givenby [31]. The relative viscosity of benzoic acid is described by [32]. Weassume the relative viscosity of SRFA to be negligible. See [33] for the rel-ative viscosity of a fulvic acid, which is seen to be a complicated functionof concentration, but with a small value. See [34] for the relative viscosityof phenylalanine solutions, which we also use for sodium phenylalanine. Wecalculate the viscosity of a mixture by first multiplying the viscosity of waterwith known relative viscosities and then using equation (73).
The density of phenylalanine and sodium phenylalanine is estimated tobe 1.227 g/cm3 using E-AIM 1 [35, 36]. The (effective) density of SRFA is1.5 g/cm3 [37]. The density of benzyl alcohol is given by [24]. The densityof benzoic acid is given by [38].
The solubility of SRFA in water is 2-3 g/l, further increasing linearlyup to 20 g/l as more SRFA is added to the system [39]. This gives usthe high and low solubilities seen in table 3. The solubilities of benzoicacid and benzyl alcohol at room temperature are 3.44 g/l [31] and 40 g/l[40] respectively. Their temperature dependence from 0◦C to 100◦C wasestimated using COSMOtherm2 [41, 42], giving the values seen in Table 3.The solubility of phenylalanine in water is given by [43], where phenylalanineis highly dissociated due to the high pH on Enceladus.
1http://www.aim.env.uea.ac.uk/aim/aim.php2http://www.cosmologic.de
31
Table 3: Solubilities of the organic compounds in mole fractionsCompound 0 ◦C 100 ◦CBenzyl alcohol 0.007 0.011Benzoic acid 0.00035 0.00225SRFA (high) 0.0006 -SRFA (low) 0.00006 -Phenylalanine 0.0004 0.002Sodium phenylalanine 0.0016 0.008
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